Interesting bounding of Analytic Functions

[Hyperbolful]

A general question I came up with and it might be trivial, but I'm not entirely sure what the answer is.

Does there exist a function analytic in the upperhalf plane that is unbounded and all of its derivatives are bounded but not identically zero?

or equivilently
does
d^n/(dz)^n(f)<M for every n imply that f is bounded

 

[Goongyae]

>Does there exist a function analytic in the upperhalf plane that is unbounded and all of its derivatives are bounded but not identically zero?

Yes.

Consider the function f(s)=s+exp(s). For Re s < 0, |f(s)| > |s+1|, so it's unbounded in the left half-plane.

It's derivatives are either 1+exp(s) or exp(s) and they are bounded for Re s < 0 with magnitudes in the range (0,2).

So the function has bounded derivatives in a half plane yet is itself unbounded there.

To make it the upper halfplane instead of the left halfplane just use f(is) instead of f(s).

 

[Goongyae]

>does d^n/(dz)^n(f)<M for every n imply that f is bounded

If f '' exists everywhere then f ' is entire. Since it is entire and bounded, it is a constant (http://en.wikipedia.org/wiki/Liouville's_theorem_(complex_analysis)). Thus f is either a constant (which is bounded) or it's As+B which is unbounded.

 

[Hyperbolful]

Yeah another counter example is sin(z)+z
I'm going to post a better thread

let f be analytic in some domain not the entire plane, and let f' be bounded

can f'' or any other derivative for that matter be unbounded?